An Exact Closed-Form Analytical Solution for Uncoupled Nonlocal Shear Elasticity with an Anti-Symmetric State-Dependent Integral Kernel: Application to Carbon Nanotube Torsion

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Abstract
While current literature heavily relies on symmetric exponential kernels within the stress-driven nonlocal integral framework, these models are strictly limited to normal bending fields and exhibit a total architectural vacancy in describing independent shear mechanics. This paper establishes the first exact analytical, closed-form constitutive model for uncoupled nonlocal shear deformation using an anti-symmetric state-dependent integral kernel.By executing a rigorous mathematical decomposition, the spatial coordinate boundaries are isolated from the polynomial domain, transforming the integral equation into a well-posed second order ordinary differential equation. Under this anti-symmetric framework, the standard boundary layer pathologies and Dirac-delta singularity spikes that historically crippled Eringen’s strain-driven configurations are identically neutralized via cross-term cancellation at the regular singular interface.The mathematical derivations reveal a pioneering structural paradigm “Automatic Dual-Phase Mechanical Bifurcation”. It is algebraically, proved that the uncoupled nonlocal shear framework automatically toggles between microstructural softening and wave-stiffening configurations based exclusively on the spatial gradient of the applied loading field. The exact formulation is successfully, deployed to evaluate the torsional shear alignment of single-walled carbon nanotubes (SWCNTs), unlocking an accurate elastic scaling resolution that remains fully singularity-free and asymptotically collapses to classical Hookean macro-continuum mechanics as the nonlocal parameter approaches zero.

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